The hat operator is a
mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...
with various uses in different branches of science and
mathematics.
Estimated value
In
statistics, a
circumflex (ˆ), called a "hat", is used to denote an
estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...
or an estimated value. For example, in the context of
errors and residuals
In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its "true value" (not necessarily observable). The error ...
, the "hat" over the letter ε indicates an observable estimate (the residuals) of an unobservable quantity called ε (the statistical errors).
In
simple linear regression with observations of independent variable data
and dependent variable data
, and assuming a model of
, can lead to an estimated model of the form
where
is minimized via
least squares by finding optimal values of
and
for the observed data.
Hat matrix
In statistics, the hat matrix ''H'' projects the observed values y of response variable to the predicted values ŷ:
:
Cross product
In
screw theory
Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawe ...
, one use of the
hat
A hat is a head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorporate mecha ...
operator is to represent the
cross product operation. Since the cross product is a
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
, it can be represented as a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. The hat operator takes a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
and transforms it into its equivalent matrix.
:
For example, in three dimensions,
:
Unit vector
In mathematics, a unit vector in a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in
(pronounced "v-hat").
Fourier transform
The
Fourier transform of a function
is traditionally denoted by
.
Quantum mechanics
In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, the capital H occurs after you partially differentiate
Schrödinger's equation.
See also
*
Exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
*
Top-hat filter
The name Top-hat filter refers to several real-space or Fourier space filtering techniques (not to be confused with the top-hat transform). The name top-hat originates from the shape of the filter, which is a rectangle function, when viewed in ...
*
Circumflex, noting that precomposed glyphs
etter-with-circumflexdo not exist for all letters.
References
Mathematical notation
{{Algebra-stub